Eq. (REF ) generalizes the contact value theorem, originally introduced for charged membranes [1]}, [2]}, [3]}, to membranes decorated with a dipolar charge. The first line on the r.h.s. of Eq. (REF ) corresponds to the electrostatic pressure, while the second line is the standard osmotic pressure associated with the contact ion density at the membrane surface. Finally, the third line is the osmotic pressure contribution stemming from the surface dipoles. Interestingly, one sees that the dipolar pressure component involves the density contrast of the surface dipoles between the orientational configurations \(\theta =\pi \)  and \(\theta =0\) . The discrepancy between this form and the standard ion contact density stems from the fact that the dipolar cations, unable to touch the membrane surface, transmit to the latter the mechanical stress originating from their rotational kinetic energy via their tail.
